Understanding the Graph of U = k*abs(x)

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In summary: I just have to memorize them?In summary, the conversation discusses the concept of absolute value and potential energy for a function, and how it relates to motion and simple harmonic motion (SHM). The speaker is asked to come up with a velocity vs. time graph for a non-SMH motion, which is derived from the force being the negative derivative of potential with respect to position. However, the speaker is unsure how to mathematically derive the equations for the sinusoidal functions of SHM and wonders if they need to be memorized.
  • #1
Gear300
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----abs( ) refers to absolute value

The potential energy for a function is given as U = k*abs(x). The motion is not SMH because U is non-differentiable at some point, but is periodic due there being a stable equilibrium. I was asked to come up with a velocity vs. time graph for this...but I don't know how the hell they got it (I don't even know how to derive the sinousidal functions for SMH)...the graph is not a trigonometric function and looks a little like this:

\/\/\/\/\/\/\ Its sort of like a sine function, except with alternating line segments.

The t-axis runs horizontally through the middle and the v-axis runs vertically so that the velocity is 0 at t = 0. How do I get this sort of graph.
 
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  • #2
Force is the (negative) derivative of potential with respect to position. So, you'll be able to find acceleration as a function of position. Then, do some chain rule stuff and you should be able to find velocity as a function of time.

edit: oh hey, you're the guy that helped me with my (other) gauss's law post.

edit2: oh wait, that just rederives K = -U.
 
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  • #3
I did what you said with a little bit of calc, but I ended up getting a linear function for velocity. The linear function does, however, portray the velocity function over selective intervals.
Guess I'll shift the question to something else: How would I mathematically derive the equations for the sinousidal functions of Simple Harmonic Motion?
 
  • #4
It's easy to find velocity as a function of position, but I'm not sure how to find it as a function of time. But, SHM (for springs) is derived like this:

[tex]F = ma => \frac{d^2x}{dt^2} = \frac{-k}{m} x[/tex]

In other words, the second derivative of position is proportional to position. The only functions that satisfy this property are the sine and cosine functions. Solving a second-order differential equation is not simple, so introductory physics books derive it this way.
 
  • #5
oh...I don't know how to solve for second-order differentials.
 
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What is the equation for the graph U = k*abs(x)?

The equation for the graph U = k*abs(x) is an absolute value function where k is a constant value and x represents the input variable.

What is the shape of the graph of U = k*abs(x)?

The graph of U = k*abs(x) is a V-shaped graph known as a "V-curve".

What is the significance of the constant k in the equation U = k*abs(x)?

The constant k in the equation U = k*abs(x) determines the steepness of the V-curve on the graph. A larger value of k results in a steeper curve, while a smaller value of k results in a flatter curve.

How do I find the x-intercept and y-intercept of the graph U = k*abs(x)?

The x-intercept of the graph U = k*abs(x) is always 0, since the absolute value of 0 is 0. The y-intercept can be found by setting x = 0 and solving for U. This results in a y-intercept of 0.

What is the domain and range of the graph U = k*abs(x)?

The domain of the graph U = k*abs(x) is all real numbers, since any value of x can be plugged into the equation. The range depends on the value of k, but it will always include 0 as the minimum value and increase from there.

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